Solve Rational Equation 1693B3

1. **State the problem:** Solve the equation $$\frac{3x}{2x-1} - \frac{2}{5-x} = 7.$$\n\n2. **Find a common denominator:** The denominators are $2x-1$ and $5-x$. The common denominator is $$(2x-1)(5-x).$$\n\n3. **Rewrite each term with the common denominator:**\n$$\frac{3x}{2x-1} = \frac{3x(5-x)}{(2x-1)(5-x)}$$\n$$\frac{2}{5-x} = \frac{2(2x-1)}{(5-x)(2x-1)}$$\n\n4. **Rewrite the equation:**\n$$\frac{3x(5-x)}{(2x-1)(5-x)} - \frac{2(2x-1)}{(5-x)(2x-1)} = \frac{7(5-x)(2x-1)}{(5-x)(2x-1)}.$$\n\n5. **Since denominators are equal, set numerators equal:**\n$$3x(5-x) - 2(2x-1) = 7(5-x)(2x-1).$$\n\n6. **Expand each term:**\n$$3x(5-x) = 15x - 3x^2,$$\n$$2(2x-1) = 4x - 2,$$\n$$7(5-x)(2x-1) = 7[(5)(2x) - 5 - 2x^2 + x] = 7(10x - 5 - 2x^2 + x) = 7(11x - 5 - 2x^2).$$\n\n7. **Simplify right side:**\n$$7(11x - 5 - 2x^2) = 77x - 35 - 14x^2.$$\n\n8. **Rewrite equation:**\n$$15x - 3x^2 - (4x - 2) = 77x - 35 - 14x^2.$$\n\n9. **Distribute the minus sign:**\n$$15x - 3x^2 - 4x + 2 = 77x - 35 - 14x^2.$$\n\n10. **Combine like terms on left:**\n$$ (15x - 4x) - 3x^2 + 2 = 77x - 35 - 14x^2,$$\n$$11x - 3x^2 + 2 = 77x - 35 - 14x^2.$$\n\n11. **Bring all terms to one side:**\n$$11x - 3x^2 + 2 - 77x + 35 + 14x^2 = 0,$$\n$$(-3x^2 + 14x^2) + (11x - 77x) + (2 + 35) = 0,$$\n$$11x^2 - 66x + 37 = 0.$$\n\n12. **Solve quadratic equation:**\n$$11x^2 - 66x + 37 = 0.$$\n\n13. **Use quadratic formula:**\n$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$$\nwhere $a=11$, $b=-66$, $c=37$.\n\n14. **Calculate discriminant:**\n$$\Delta = (-66)^2 - 4 \times 11 \times 37 = 4356 - 1628 = 2728.$$\n\n15. **Simplify square root:**\n$$\sqrt{2728} = \sqrt{4 \times 682} = 2\sqrt{682}.$$\n\n16. **Write solutions:**\n$$x = \frac{66 \pm 2\sqrt{682}}{22} = \frac{66}{22} \pm \frac{2\sqrt{682}}{22} = 3 \pm \frac{\sqrt{682}}{11}.$$\n\n17. **Final answer:**\n$$x = 3 + \frac{\sqrt{682}}{11} \quad \text{or} \quad x = 3 - \frac{\sqrt{682}}{11}.$$